Introduction to Linear Inequalities

When one expression is given to be greater than or less than another expression, we have an inequation.For example, consider:

\[2x + 3 > 7\]

This is an example of an inequation in one variable. The solution to this inequation will be the set of all values of x for which this inequationis satisfied, that is, the left side is greater than the right side. The solution in this case is simple to evaluate:

Thus, all values of x greater than 2 will satisfy this inequality, because for all values of x greater than 2, the term \(2x + 3\) will be greater than 7.

Now, consider the inequation

\[x + 2y > 3\]

This is an example of an inequation in two variables. The solution to this inequation will be the set of all pairs of values for x and y such that the expression \(x + 2y\) is greater than 3.For example, one possible solution is \(x =3,\;y = 1,\) because

\[\left( 3 \right) +2\left( 1 \right) = 5 > 3\]

Note that this is just one possible solution. The solution to the inequation as a whole will be all pairs of values which satisfy the inequation.

The two examples considered above have strict inequalities: this means that the two sides can never be equal. However, we can also have inequations which don’t have strict inequalities. For example,

\[\begin{array}{l} -3x + 2 \ge 5\\x - 4y \le \sqrt 2 \end{array}\]

are linear inequations which are not strict, because the two sides in each inequation can also be equal.

As we have seen, a linear inequation in one variable involves an inequality between two linear expressions, or between a linear expression and a constant, where there is only one variable involved. Some examples:

Now, we want to represent this solution set on a number line. Thus, we simply highlight that part of the number line lying to the left of 2:

We see that any number lying on the red part of the number line will satisfy this inequation, and so it is a part of the solution set for this inequation. Note that we have drawn a solid dot exactly at the point 2. This is to indicate that 2 is also a part of the solution set.

Any point lying on the red part of the number line will satisfy this inequation. Note that in this case, we have drawn a hollow dot at the point 3. This is to indicate that 3 is not a part of the solution set (this is because the given inequation had a strict inequality).

Is this solution correct? No, it is not! Let us try to understand why. Let us pick an arbitrary number in this solution set, say \(x = 2,\) and see whether it satisfies the original inequation. We have

\[ - 2\left( 2 \right)+ 3 = - 4 + 3 = - 1\]

This is not greater than 5, and thus x equal to 2 does not satisfy the original inequation, which means that the solution set \(x > - 1\) is incorrect. Where did we go wrong?

No! This is incorrect. If –xis greater than 1, this cannot mean that x is greater than –1. In fact,what this will actually mean is that x is less than –1. Only then can the negative of x be greater than 1. Take your time and reflect on this argument in detail.

Thus, when we multiplied the inequation by –1 on both sides, we should actually have reversed the direction of the inequality: